We may regard the left-hand side of the equation of thecurve as a quadratic polynomial in $x$. If $D(y)$ is its discriminant (with respect to $x$), then $D(y)\ge 0$ iff there exists a point with the second coordinate $y$ on the curve. Solve this inequality for $y$ and check whether its minimal solution is negative:)))

I am not sure I understand the question. A point ON the $x$ axis is given by $y=0,$ so $a_{11} x^2 + 2 a_{13} x + a_{33}=0.$ That IS a parabola, and so we know what the solutions are (if any). If if the discriminant is positive, you are golden. If the discriminant is 0, pick a random value of $x$ ($0$ is easiest, but that might be the solution to your quadric), and check if $y$ is positive. If the discriminant is negative, take $x = 0$ and see what the solutions are to the resulting quadric in $y.$